Another special relativity related "paradox"

[sphyrch]

Full disclosure: I have asked this question on stackexchange too, but I think I didn't frame my question properly there, which probably led to misunderstandings and complicated answers. Plus some comments there led me to refine the question a bit so I hope it's in a good state now.

I am new to special relativity and was watching scienceclic's video on it, when I thought of this scenario:

A rocket is moving w.r.t. the earth. Rocket guy is standing in the middle of the rocket and earth guy is watching the rocket from earth. Rocket guy shines a laser in both directions - the effect of the laser is that the part of the rocket it hits will turn red. So now earth guy will see that the light will hit the rear end first, and then the front end.

This is a spacetime diagram I tried to make for earth guy. I've labeled worldlines for rear and front ends of the rocket, and "mid" worldlines is that of rocket guy. My confusion is this - at a point in time between t1 and t2, earth guy can take a picture of the rocket - and the photo will show that the rear end of the rocket is red and the front end is normal. Rocket guy will swear that there was never any color difference between the rocket ends, since for him the lasers reached simultaneously. But earth guy can show him the pic and contradict.

Obviously I'm making a mistake or wrong assumption somewhere, but can't figure it out. Will be great if I can get help on this!

 

[A.T.]

earth guy can take a picture of the rocket - and the photo will show that the rear end of the rocket is red and the front end is normal. Rocket guy will swear that there was never any color difference between the rocket ends, since for him the lasers reached simultaneously. But earth guy can show him the pic and contradict.

Taking a picture involves light traveling from the rocket to the camera, which is affected by relativity of simultaneity and signal delay. Both frames will agree what the camera shows, when they analyze the travel of the lasers and of the colored light to the camera.

 

[PeterDonis]

at a point in time between t1 and t2, earth guy can take a picture of the rocket - and the photo will show that the rear end of the rocket is red and the front end is normal

This statement is actually not correct as you state it! Think about it: the light that is entering the camera has to travel from the front and rear ends of the rocket. So if you want to pick out the times along Earth's worldline where a camera located there will take a picture showing the rear end of the rocket as red and the front end as not red, you need to draw on your diagram the appropriate worldlines of the light rays (45 degree lines up and to the left) from the events you have labeled "rear turns red" and "front turns red", and the points of intersection of those worldlines with the Earth worldline will label the times in between which a camera located on Earth will take the picture you describe. Obviously the first time must be later than t1, because at t1 light from the event "rear turns red" cannot possibly have reached Earth; and the second time must be (quite a bit more) later than t2 for a similar reason.

This point does not change the response to your other statement, however:

Rocket guy will swear that there was never any color difference between the rocket ends, since for him the lasers reached simultaneously. But earth guy can show him the pic and contradict.

No, Earth guy's picture does not contradict, because Earth's camera and rocket guy's camera are at different events when they take the pictures in question. Rocket guy's camera is traveling on rocket guy's worldline, and at any event on this worldline, light rays coming in from the front and rear of the rocket will show no color difference. So there is no contradiction, just the fact that cameras at different events in spacetime can take different pictures.

 

[sphyrch]

Taking a picture involves light traveling from the rocket to the camera, which is affected by relativity of simultaneity and signal delay. Both frames will agree what the camera shows, when they analyze the travel of the lasers and of the colored light to the camera.

Thanks I see what you mean. I attempted to factor that into the spacetime diagram like so:

Please note that blue and red worldlines are supposed to overlap - I've shown them very slightly apart for clarity. Let's say red light travels from the "rear turns red" event to earth guy's camera, and reaches the camera at time t3. At the same time, the light from the front end of the rocket that reaches camera at t3, will have come from an event at which the front end was still normal.

So isn't the camera going to see normal front and red rear?

 

[PeterDonis]

I attempted to factor that into the spacetime diagram like so

Yes, the blue line represents the worldline of a light ray (or actually two light rays, since both the front and the rear are included in the "image") reflecting a red rear (because it's emitted from the rear worldline just an instant after the "rear turns red" event) and a normal front (because it's emitted from the front worldline long before the "front turns red" event).

isn't the camera going to see normal front and red rear?

Yes. See above.

But now try drawing light rays from front and rear going to Rocket guy's camera, which is on Rocket guy's worldline, not Earth's worldline. You will find that no matter what you do, it will be impossible to have light rays arriving at the same point on Rocket guy's worldline but showing different colors for front and rear. And, as I said before, this is not a problem! It's just the natural consequence of having the cameras at different events in spacetime.

 

[sphyrch]

No, Earth guy's picture does not contradict, because Earth's camera and rocket guy's camera are at different events when they take the pictures in question. Rocket guy's camera is traveling on rocket guy's worldline, and at any event on this worldline, light rays coming in from the front and rear of the rocket will show no color difference. So there is no contradiction, just the fact that cameras at different events in spacetime can take different pictures.

This I didn't expect. In that case no contradiction to begin with.

The reason I was struggling was - I was thinking what if the laser simultaneously hitting the ends of the rocket causes some material physical change - like some device that short circuits if both ends are hit simultaneously, or some object whose physical property changes if both ends are hit simultaneously.

In that case there could be a contradiction since the rocket guy can always stop on earth and show the short-circuited device or changed object to earth guy. The earth guy would be surprised since from his pov the rocket ends weren't hit simultaneously and the physical change to the device/object shouldn't have happened.

But now that I think about it, I can't just ignore the mechanism by which these physical changes happen, as if they happen by magic. In other words, earth guy wouldn't even agree with the "rule" that the physical change happens when both ends are hit simultaneously. He'll just tell rocket guy, "so what? i know that the device would short-circuit if the rear end was hit by the laser first"

Is the above okay?

 

[PeterDonis]

I can't just ignore the mechanism by which these physical changes happen, as if they happen by magic.

Exactly. Whatever "physical change" happens has to happen because of some mechanism, and has to happen because of some actual physical trigger. For example, the changing of color of the rocket in your scenario is a physical change, and it happens because of the light rays striking the ends of the rocket. Those strikes are physical events in spacetime. They could just as easily trigger explosions, or opening or closing of doors, or whatever physical change you like. But the change will be at some event in spacetime; it won't just magically "happen" without being anchored in the spacetime geometry. It's that anchoring in the spacetime geometry that ensures that no actual contradictions arise.

earth guy wouldn't even agree with the "rule" that the physical change happens when both ends are hit simultaneously

If the "rule" is just stated as magic, no, of course not. But if the "rule" is, for example, "the ends of the rocket change color when the light rays emitted by Rocket guy strike them", then Earth guy would have to agree with that rule--because it's not magic, it's based on invariant physical events that are anchored in the spacetime geometry, and that anchoring is the same for everyone. Earth guy would not agree on the coordinate times at which these events happen, but coordinate times are just abstractions. The important point is that Earth guy and rocket guy have to agree on what actually happens. As long as the "rule" for what happens is stated in terms of invariant physical events, not abstractions like "simultaneous" or coordinate time, there's no problem.

 

[PeterDonis]

some device that short circuits if both ends are hit simultaneously

But the ends are separated in space, so a "short circuit" can't happen instantly; it takes time for the circuit to be completed. Or, in spacetime terms, suppose that at the events when the light strikes each end of the rocket, a switch is closed, and there are wires connecting the ends and a battery in the circuit. The closing of the switches, in itself, does not instantly cause current to flow everywhere in the circuit. It will take time--at best, current can only flow from the battery around the circuit at the speed of light. So you would have to work out on the spacetime diagram when current actually started flowing at each point in the circuit by taking that into account.

 

[sphyrch]

Exactly. Whatever "physical change" happens has to happen because of some mechanism, and has to happen because of some actual physical trigger. For example, the changing of color of the rocket in your scenario is a physical change, and it happens because of the light rays striking the ends of the rocket. Those strikes are physical events in spacetime. They could just as easily trigger explosions, or opening or closing of doors, or whatever physical change you like. But the change will be at some event in spacetime; it won't just magically "happen" without being anchored in the spacetime geometry. It's that anchoring in the spacetime geometry that ensures that no actual contradictions arise.If the "rule" is just stated as magic, no, of course not. But if the "rule" is, for example, "the ends of the rocket change color when the light rays emitted by Rocket guy strike them", then Earth guy would have to agree with that rule--because it's not magic, it's based on invariant physical events that are anchored in the spacetime geometry, and that anchoring is the same for everyone. Earth guy would not agree on the coordinate times at which these events happen, but coordinate times are just abstractions. The important point is that Earth guy and rocket guy have to agree on what actually happens. As long as the "rule" for what happens is stated in terms of invariant physical events, not abstractions like "simultaneous" or coordinate time, there's no problem.

Thanks for the detailed explanation. So basically the statement that "the device short-circuits if both ends of the rocket are hit at the same time" is not a "rule" since it relies on coordinate time

 

[PeterDonis]

So basically the statement that "the device short-circuits if both ends of the rocket are hit at the same time" is not a "rule" since it relies on coordinate time

Yes.

 

[FactChecker]

The OP makes the common mistake of talking about "seeing" and "photograph" which requires that light travel back from the event to an observer at a different location. That conceptually complicates the problem.

Assume that each inertial reference frame has observers at every location who can record the time of any event in their IRF location, and the clocks in that IRF have all been synchronized. Then the spaceship reference frame will record that the ends turned red at the same time according to clocks synchronized in the spaceship IRF. The Earth IRF will record that the ends turned red at different times according to clocks synchronized in the Earth IRF.
This can be summarized by saying that simultaneity is relative.

 

[Halc]

For the record, this sort of thing is usually done with a train, not a rocket.
Einstein had some pretty fast trains when he was a kid.

at a point in time between t1 and t2, earth guy can take a picture of the rocket - and the photo will show that the rear end of the rocket is red and the front end is normal.

This statement is actually not correct as you state it!

Sure it is. Put the camera well off to the side in the y direction, even with where the midpoint will be when the photo is taken. The light from when the rocket was at t=[somewhere betweeen t1 & t2] will show the rear red and the front not, two events simultaneous in the 'Earth' frame.

The explanation for this has already been explained above. It is simply an illustration of relativity of simultaneity, and the photo takes a picture if a different pair of events simultaneous in the camera frame (red and not) than the pair of events simultaneous in the rocket frame (the laser first hitting each end).

 

[PeterDonis]

Sure it is.

No, it isn't. Read carefully. The claim I was referring to was the claim that between times t1 and t2 the Earth camera will see the rear end of the rocket red and the front end normal. That claim is not correct because the times are wrong. I explained why.

Put the camera well off to the side in the y direction

Then the camera will not be the Earth guy's camera, and the statement I was responding to was about the Earth guy's camera. Again, read carefully.

 

[A.T.]

So there is no contradiction, just the fact that cameras at different events in spacetime can take different pictures.

This I didn't expect.

Note that even without relative motion, camera recordings can disagree on whether some events happened simultaneously or not, simply due to finite signal speed and different distances to the camera.

 

[A.T.]

The OP makes the common mistake of talking about "seeing" and "photograph" which requires that light travel back from the event to an observer at a different location.

I think this is due to the common use of "observer" as a synonym for "reference frame".

 

[FactChecker]

I think that the OP scenario is exactly the setup that is called Einstein synchronization convention of clocks. The light from the center point goes to the ends, which turn red immediately and the red light travels back to the (moving wrt Earth) spaceship center. The observer on the spaceship sees the red at the same time and says that the light beams hit the ends "simultaneously", i.e. the clocks at the ends are Einstein synchronized. The Earth observer sees the red at different times and concludes that the spaceship clocks are not correctly synchronized for his Earth IRF.
I think that understanding the concepts of Einstein synchronization of clocks and of the relativity of simultaneity would answer the question in the OP.
There are still problems with the Einstein synchronization convention but I don't think that the OP scenario has those issues.